Ordinal Arithmetic Calculator – Addition, Multiplication & Exponentiation

Calculate with ordinals

Enter expressions using ω (or omega), integer exponents, coefficients, and parentheses for compound exponents. Terms must be written in descending Cantor normal form.

Try an example:

ω + 1 1 + ω ω × 2 2 × ω ω³ (ω² + 1)²

Supported ordinal notation

The calculator accepts Cantor normal form expressions built from powers of ω with positive integer coefficients, together with finite ordinal terms.

ω^(ω+1)*2 + ω^3*4 + ω*5 + 7

Notation rules:

ω or omega represents the first infinite ordinal.
ω^3 represents ω raised to the third power.
ω^2*3 represents three copies of the term ω² in Cantor normal form.
ω + 5 represents the ordinal sum of ω and 5.
ω^(ω+1) uses parentheses for a compound exponent.
0 represents the zero ordinal.

Important: Write terms in descending order of their exponents. For example, ω^2 + ω + 3 is valid, while ω + ω^2 + 3 is not in Cantor normal form.

How ordinal arithmetic works

Ordinal addition

Ordinal addition describes placing one well-ordered set after another. It is not commutative: changing the order of the operands can change the result.

1 + ω = ω

ω + 1 = ω + 1

The first expression has order type ω because adding one element before an infinite sequence does not change its order type. Adding one element after that sequence produces a larger ordinal.

Ordinal multiplication

Ordinal multiplication represents repeated ordinal addition, with the order of the factors being significant.

ω × 2 = ω + ω = ω·2

2 × ω = ω

Thus, ordinal multiplication is not commutative either.

Ordinal exponentiation

Ordinal exponentiation extends repeated ordinal multiplication. For a finite exponent n, αⁿ can be evaluated through n successive ordinal multiplications, with α⁰ = 1 under the convention used here.

ω² = ω × ω

ω³ = ω × ω × ω

This calculator evaluates finite ordinal exponents exactly within its resource limits. It does not evaluate exponentiation with an infinite ordinal exponent.

Cantor normal form

Every ordinal below ε₀ has a unique representation in Cantor normal form:

ω^α₁·c₁ + ω^α₂·c₂ + ... + ω^αₙ·cₙ

Here:

Each exponent α is itself an ordinal.
Each coefficient c is a positive finite integer.
The exponents are arranged in strictly descending order.
The finite ordinal is represented by a final constant term, when present.

The ordinal ε₀ is the limit of the sequence ω, ω^ω, ω^(ω^ω), and so on. It is the first ordinal satisfying ω^ε₀ = ε₀. The recursive Cantor normal forms supported here describe ordinals below that boundary.

Calculator limitations

Inputs use Cantor normal form with nonnegative integer coefficients and finite integer exponents at the outermost level.
Compound exponents can be entered in parentheses, such as ω^(ω+1).
Ordinal addition and multiplication are evaluated using exact Cantor normal form rules.
Exponentiation is supported when the exponent is finite.
Exponentiation with an infinite ordinal exponent is not implemented.
Very large expressions may be rejected to protect server resources.

This is an exact calculator for its supported operations and input range, not a general-purpose implementation of every ordinal operation.

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Frequently asked questions

What is an ordinal number?

An ordinal describes the order type of a well-ordered set. Finite ordinals describe ordinary counting positions, while infinite ordinals describe well-ordered structures beyond the finite numbers.

What is Cantor normal form?

Cantor normal form expresses an ordinal as a finite sum of decreasing powers of ω with positive integer coefficients. It provides a canonical representation for every ordinal below ε₀.

Is ordinal addition commutative?

No. For example, 1 + ω = ω, whereas ω + 1 is strictly larger than ω. The order of the operands matters.

Is ordinal multiplication commutative?

No. For example, 2 × ω = ω, but ω × 2 = ω·2.

Can I calculate with ε₀?

This calculator represents ordinals below ε₀ using recursive Cantor normal form. It does not accept ε₀ itself as an ordinary input expression.

Does the calculator support infinite exponents?

Infinite exponents can appear inside Cantor normal form terms, but the exponentiation operation currently requires its second operand to be finite. It reports an error rather than returning an unsupported or approximate answer.